Primitive of Reciprocal of x squared plus a squared/Arccotangent Form
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Theorem
- $\ds \int \frac {\d x} {x^2 + a^2} = -\frac 1 a \arccot \frac x a + C$
where $a$ is a non-zero constant.
Proof
\(\ds \int \frac {\d x} {x^2 + a^2}\) | \(=\) | \(\ds \frac 1 a \int \frac {\d t} {t^2 + 1}\) | Substitution of $x \to a t$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\d t} {\paren {1 + i t} \paren {1 - i t} }\) | factoring | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \paren {\int \frac {\d t} {1 + i t} + \int \frac {\d t} {1 - i t} }\) | Definition of Partial Fractions Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \paren {i \map \ln {1 - i t} - i \map \ln {1 + i t} } + C\) | Primitive of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac i {2 a} \map \ln {\frac {1 - i t} {1 + i t} } + C\) | Sum of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac i {2 a} \map \ln {\frac {1 + i t} {1 - i t} } + C\) | Logarithm of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 a \arccot \frac x a + C\) | Arccotangent Logarithmic Formulation and substituting back $t \to \dfrac x a$ |
$\blacksquare$
Sources
- 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Appendix $\text I$: Table of Indefinite Integrals $12$.
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $27$
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals